Problem: The $n\text{th}$ partial sum of the series $\sum\limits_{n=1}^{\infty }{{{a}_{n}}}$ is given by ${{S}_{n}}=\frac{n-1}{n+1}$. Write a rule for ${{a}_{n}}$.
Explanation: The $~n\text{th}~$ term of the series is the difference between successive partial sums. That is, ${{a}_{n}}={{S}_{n}}-{{S}_{n-1}}\,$. $\begin{aligned} {{S}_{n}}-{{S}_{n-1}}&=\frac{n-1}{n+1}-\frac{n-2}{n} \\\\ &=\frac{n(n-1)}{n(n+1)}-\frac{(n+1)(n-2)}{n(n+1)} \\\\ &=\frac{2}{n\left( n+1 \right)} \end{aligned}$